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Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations

Published online by Cambridge University Press:  14 November 2011

Michel Chipot  and
Lawrence C. Evans
Michel Chipot
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
Lawrence C. Evans
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
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Synopsis

We demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 291 - 303
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Giaquinta, M.. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems (Princeton: Princeton University Press, 1983).Google Scholar
2Giusti, E. and Miranda, M.. Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasilineari. Arch. Rational Mech. Anal. 31 (1968), 173–184.CrossRefGoogle Scholar
3Kohn, R. and Strang, G.. Optimal design and relaxation of variational problems. Comm. Pure Appl. Math, (to appear).Google Scholar